For Ref. [1]}, both the first, \(\tilde{\epsilon }_1\) , and the third, \(\tilde{\epsilon }_3\) , harmonics of \(\delta U\)  are sizable (see Table REF ). The wavevectors of the third harmonic are \(\pm 2 g_1\) , \(\pm 2 g_2\) , and \(\pm 2(g_1 + g_2)\) . Following the argument in the above paragraph, we therefore expect that \(H_{inter}\)  should include the terms with the momentum transfer of all \(q\) s in the first 6 shells. Indeed, as demonstrated numerically in the Fig. REF (b), we achieve the accuracy of \(0.8\) meV around \(\Gamma \)  with six \(q\)  shells.
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